Download Use of Generalized Extreme Value distribution to model the

Use of Generalized Extreme Value distribution to model the maximums of the field
inside a reverberation chamber
Gérard Orjubin1, Man-Fai Wong2
1
Université Française d’Egypte, Egypte. Email: [email protected]
2
France Télécom R&D, RESA/SAFE. 38-40, rue du Général Leclerc, 92794 Issy Moulineaux, France.
E-mail: [email protected]
Abstract
The Generalized Extreme Value (GEV) distribution is used to model the electric field inside a Reverberation
Chamber (RC). GEV distribution is particularly indicated for this application where the maximums of the field are
studied, as it specifies the theoretical distribution of the maximums according to the Fisher-Tippett theorem. In a
first part of this paper we present the Fisher-Tippett theorem and the GEV distribution. Then the three parameters of
the GEV distribution are estimated from measured maximum values of the field in a small RC equipped with two
stirrers.
1. Introduction
Reverberation chambers (RCs) are tools used for electromagnetic compatibility tests. An RC is a metallic
enclosure equipped with a mechanical stirrer that modifies the geometrical configuration. For each stirrer position,
the electric field is composed of eigenmodes, each of these standing waves presenting nodes and antinodes. The
nonuniformity of the resulting field can be statistically suppressed, considering the maximum of the field for a stirrer
rotation. The probabilistic description of the field has been proposed by Hill [1] for the electrically large (or
overmoded) RC and has been validated [2]: The rectangular electric field components E R follow a Rayleigh
distribution function (DF).
A useful application of this model is the determination of the field uniformity, as defined in the 61000-4-21
norm [3]. First, E R max , which are the maximums of E R within a stirrer displacement, are collected, and then the
uniformity estimator S is defined as the normalized standard deviation of E R max . The statistics of E R max can be
predicted by the use of extreme order statistics as soon as the E R distribution, which is also called the parent DF, is
known.
Because the parent distribution is not always known (such as for the undermoded RC operated at low
frequencies), it can be helpfull to apply the Fisher-Tippett theorem that gives the three asymptotic possible DFs of
the maximums. This technique has been applied in [4] to model the extreme power consumption in a VLSI circuit.
As an extension, we have adopted in [5] the generalized extreme value (GEV) DF that encompasses these three
limiting DFs. The results of [5] were related to a modeled RC, the ones of this paper concern a small RC whose
volume is 1.3m3.
The outline of this paper is as follows. In Section 2, the results of the classical use of extreme order
statistics in relation to an RC are recalled, followed by presentations of both the Fisher–Tippett theorem and GEV
DF. In Section 3, measurement are performed and the maximums are modeled by GEV.
2. Theoretical background
2.1 Uniformity test derived from the study of the maximums
The calibration procedure [3], which leads to an estimation of the RC field uniformity, is based on the
determination of the maximum field over a stirrer displacement. To better understand the consequence of this choice
and introduce our modeling, we recall how the uniformity estimator S is defined. Then, the well known theoretical
model for the overmoded RC, making use of extreme order statistics, is presented. To begin with, the electric field
magnitude is determined at eight points located at the working volume corners. For each rectangular component E R
of a given point, E R max corresponds to the maximum value over N1 independent stirrer positions. From these 8×3
rectangular components E R max the field uniformity is evaluated by calculating the mean value < E R max > and the
empirical standard deviation σ ER max . Then, the normalized dispersion S is defined as
σ ER max ⎞
⎛
⎟.
S = 20 log10 ⎜⎜1 +
⎟
⎝ < E R max > ⎠
(1)
The 61000-4-21 norm states that uniformity is achieved if S is inferior to 3dB.
The theoretical determination of S is based on the use of extreme order statistics, whose principle is the
following: Given F, the hypothesized cumulated density function (cdf) of the field, or parent DF, it is easy to show
[2, Sec. 3.2.6] that the cdf G of the maximum among N1 values is
G = F N1 .
(2)
For the overmoded RC, parent DFs are well known: E R and E R2 follow Rayleigh and exponential
distributions, respectively. Making use of (2), it is then possible to express the mean value and the dispersion of G as
functions of N1. If the aim is to model the maximum of E R , these expressions are in [2, Tab. 4]: For instance N1 =
20 independent samples lead to S = 1.37 dB. In the case of the undermoded RC, this method cannot be applied
since no modeling of the parent distribution is currently accepted. To circumvent the E R modeling issues, we use an
asymptotic expression of (2) that is valid for both overmoded and undermoded RCs.
2.2 Fisher-Tippett theorem
In the same way that the central limit theorem gives the asymptotic probability density function of sample
average, the Fisher–Tippett theorem [6] indicates the possible limiting cdfs G∞ of normalized sample maximums
that are independent and identically distributed. This theorem states that G∞ can be only one among the three types
[7, pp. 51–52]:
x≤0
⎧ 0
(3a)
G1 ( x) = ⎨
if
−α
x>0
⎩exp − x
[
]
[
⎧exp − (− x )α
G 2 ( x) = ⎨
1
⎩
[
G3 ( x) = exp − e − x
]
]
if
x≤0
x>0
∀x ∈ ℜ
(3b)
(3c)
where α > 0 is the tail index. Equations (3a)–(3c) are the asymptotic cdfs of the maximums of X. Conditions
expressed in [7, pp. 51–52] allow a determination of the type of limiting distribution for usual DFs:
1) G1 is Fréchet DF. Convergence towards G1 corresponds to “fat tail” parent DF, e.g., Pareto or Cauchy DF. The
terminology “fat tail” comes from the visual inspection of the cdf plot: If the probability of extreme values is
particularly high, the cdf converges slowly towards the asymptotic value 1, yielding a fat tail.
2) G2 is the reverse Weibull DF. Convergence towards G2 corresponds to thin-tailed parent DF with a finite upper
end point such as beta or uniform DF.
3) G3 is Gumbel DF or simply extreme value DF, as it is the standard case for the following parent DFs: normal, log
normal, gamma, Weibull, and Rayleigh.
Since the values of the field inside the RC are right bounded, one could presume that the case 2) is suited to
model our application. However, it has been pointed in [8] that case 3) often models better the empirical data.
2.3 Generalized extreme value distribution
The GEV DF combines into a single form the three limiting DFs [(3a)–(3c)]. The definition3 [9, p. 4] of the GEV cdf
for maximum values is
GGEV
GGEV
1
− ⎤
⎡
k
x
m
−
⎞
⎛
( x) = exp ⎢− ⎜1 + k
⎟ ⎥, k ≠0
⎢ ⎝
s ⎠ ⎥
⎣
⎦
⎡
⎛ x − m ⎞⎤
( x) = exp ⎢− exp ⎜1 −
⎟ , k =0
s ⎠⎥⎦
⎝
⎣
(4a)
(4b)
where k is the shape parameter. Equations (3a)–(3c) can be obtained with the following values of (k, s, m): Fréchet
(1/α, 1/α, 1), Weibull (−1/α, 1/α,−1), and Gumbel (0, 1, −1). With linear rescaling [5], the correspondence between
GEV and Fisher–Tippett types is given in Table I.
TABLE I: CORRESPONDENCE BETWEEN GEV AND FISHER TIPPETT TYPES
GEV shape parameter
k>0
k<0
k=0
Fisher-Tippett type
Fréchet
Weibull
Gumbel
Domain
x > m - s/k
x < m - s/k
-∞ < x < +∞
As an example, GEV cdfs are graphed in Fig. 1 for (s, m) = (6, 10) and k taking the values −0.2, 0, and 0.2.
1
0.9
0.8
cdf of GGEV
0.7
0.6
0.5
0.4
0.3
Frechet
Gumbel
Weibull
0.2
0.1
0
-10
0
10
20
30
40
50
x
Fig. 1. GEV cdfs for (s, m) = (6, 10) and k taking the values -0.2, 0 and 0.2.
Fig. 1 shows the critical importance of the sign of k for the type of GEV DF: Gumbel DF is unbounded, Fréchet DF
has a lower end point (x > −20), and Weibull DF has an upper end point (x < 40). Besides this property, the most
interesting fact concerns the tail: Fréchet (resp.Weibull) corresponds to a fatter (resp. finner) tail.
3. Experiment
The measurements are performed within the France Télécom facilities with the Bluetest® RC whose
dimensions are 0.80 m × 1.05 m ×1.60 m. We operate at f = 2.45 GHz where the RC is supposed to be overmoded
(the manufacturer recommends 870 MHz as the minimum frequency). The CW power (-4.84 dBm) is transmitted by
a broadband dipole antenna located at the higher part of the RC. Two stirrers are translated in a synchronous way
yielding 20 independent positions. The three components of the electric field are measured with a 3D probe (PMM
EP 330) placed at 27 positions, with 21 cm and 18 cm horizontal steps and 24 cm vertical step. These distances
insure the independence of the data as the minimum distance is larger than a quarter of the wavelength (3.1 cm). The
maximums of the E R component over the 20 geometrical configurations (associated to the stirrer displacement) are
determined and the uniformity estimator (1) is found to be S = 1.58 dB for the 27 × 3 data set. Comparing to the
theoretical 1.37 dB derived for overmoded RCs we see that 2.45 GHz is a frequency at which the uniformity is
attained but some deviation may exist from the asymptotic case.
The GEV parameters are estimated by Maximum of Likelihood method that is reliable for medium sized
data: this is the reason why we used 27 measurement points instead of 8 as advised by the norm. One can see in Fig.
2 that the empirical cumulated density function is well fitted by GEV cdf.
Empirical and GEV estimated cdf (ML method)
1
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
6
7
8
9
10
11
12
x=ERmax (V/m)
13
14
15
16
Fig. 2. GEV cdf and empirical cdf of the maximums of E R .
The estimated GEV parameters are (k, s, m) = (0.03, 1.51, 8.47). The vanishing k value indicates that the
distribution is of Gumbel type, in compliance with the Fisher-Tippett theorem as the parent distribution is Rayleigh
for this fairly overmoded RC. Finally we can take advantage of the knowledge of the theoretical distribution to
determine the quantile xp for a given probability p. Inverting (4a) it comes
xp =m −
( (
))
s
−k
1 − − log(GGEV )
, k ≠ 0,
k
with GGEV = 1 − p.
This GEV distribution has no upper bound as k ≥ 0. At a 95% confidence level, the maximum value of the electric
field is 12.7 V/m. This technique can be of a great interest for some applications [10].
4. References
1. D. A. Hill, “Electromagnetic theory of reverberation chamber,” Nat. Inst. Standards Technol., Boulder, CO, NIST
Tech. Note 1506, 1998.
2. J. M. Ladbury, G. H. Koepke, and D. G. Camell, “Evaluation of the NASA Langley Research Center ModeStirred Chamber Facility,” Nat. Inst. Standards Technol., Boulder, CO, NIST Tech. Note 1508, 1999.
3. IEC 61000-4-21 Electromagnetic Compatibility: Reverberation Chamber Test Methods. Intern. Electrotech.
Commission (IEC), Genève, Switzerland, Aug. 2003.
4. Q. Wu, Q. Qiu, and M. Pedram, “Estimation of peak power dissipation in VLSI circuits using the limiting
distributions of extreme order statistics,” IEEE Trans. Comput. Aided Des., vol. 20, no. 8, pp. 942–956, Aug.
2001.
5. G. Orjubin, “Maximum field inside a reverberation chamber modeled by the generalized extreme value
distribution,” IEEE Trans. Electromag. Compat., vol. 49, no. 1, pp. 104–113, Feb. 2007.
6. R. A. Fisher and L. H. C. Tippett, “Limiting forms of frequency distributions of the largest or smallest member of
a sample,” Proc. Cambridge Philosophical Soc., pp. 180–190, vol. 24, 1928.
7. J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1987.
8. J. Galambos and N. Macri, “Classical extreme value model and prediction of extreme winds,” J. Struct. Eng., vol.
125, no. 7, pp. 792–794, Jul. 1999.
9. S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications, World Scientific, Singapore,
2001.
10. T. Koneval, A. C. Marvin, J. F. Dawson, and M. P. Robinson, “A statistical model to estimate an upper bound on
the probability of failure of a system installed on an irradiated vehicle,” IEEE Trans. Electromagn. Compat., vol.
49, no. 3, pp. 840–848, Nov. 2007.
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